[en] Evolutionary game theory is the application of game theory to evolving populations in different contexts as biology or economics. It defines a theoretical framework of contests and strategies into which Darwinian competition can be modelled. In this work, we will present the main concepts of this field from a mathematical perspective, being the evolutionary stable strategies (ESS) and the replicator dynamics systems the most important concepts. Firstly, we present the basic concepts of game theory and qualitative theory of differential equations that will be used throughout the work. We also study how to represent plannar differential systems using the Poincaré compactification. Then, we develop a theoretical study of the ESS and explain the different type of games where this concept can be applied, player versus environment games and player versus player games. Finally, we explain the replicator dynamics and weaker concepts of evolution than ESS. All the sections will have different examples in order to help the reader understand and apply the theory.

Treballs Finals del Doble Grau d'Administració i Direcció d'Empreses i de Matemàtiques, Facultat d'Economia i Empresa i Facultat de Matemàtiques i Informàtica, Universitat de Barcelona, Curs: 2020-2021, Tutors: Xavier Jarque i Ribera i Josep Maria Izquierdo Aznar